Tunable Aharonov-Bohm-like cages for quantum walks
TTunable Aharonov-Bohm-like cages for quantum walks
Hugo Perrin, ∗ Jean-No¨el Fuchs, † and R´emy Mosseri ‡ Sorbonne Universit´e, CNRS, Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, LPTMC, 75005 Paris, France (Dated: July 10, 2020)Aharonov-Bohm cages correspond to an extreme confinement for two-dimensional tight-bindingelectrons in a transverse magnetic field. When the dimensionless magnetic flux per plaquette f equals a critical value f c = 1 /
2, a destructive interference forbids the particle to diffuse away from asmall cluster. The corresponding energy levels pinch into a set of highly degenerate discrete levels as f → f c . We show here that cages also occur for discrete-time quantum walks on either the diamondchain or the T tiling but require specific coin operators. The corresponding quasi-energies versus f result in a Floquet-Hofstadter butterfly displaying pinching near a critical flux f c and that may betuned away from 1/2. The spatial extension of the associated cages can also be engineered. I. INTRODUCTION
Two-dimensional (2d) electronic systems in a perpen-dicular magnetic field have been of special interest incondensed matter physics. They lead to various subtleeffects such as the measured integer [1] and fractional [2]quantum Hall states or the predicted self-similar Hofs-tadter butterfly [3] describing the structure of energy lev-els for tight-binding (TB) electrons versus the magneticflux. The possibility of generating “artificial” magneticfields acting on cold atoms assemblies opens the way todifferent types of experiments in that direction [4].About 20 years ago, an extreme localization effect wasproposed, which occurs for TB models in certain periodiclattices, such as the 2d T (or dice) rhombus tiling [5]or the diamond chain (DC) [6], at half a magnetic fluxquantum per plaquette. These so-called Aharonov-Bohm(AB) cages are due to a complete destructive interfer-ence preventing the particle to escape from finite clusters.Generic periodic structures, like the infinite square lat-tice studied in [3], lead to energy bands for rational fluxes(measured in units of the flux quantum), and more com-plex density of states for irrational fluxes. In contrast,the T butterfly displays the surprising feature that thedensity of states pinches near a half flux quantum, lead-ing to an energy spectrum consisting of three highly de-generate energy levels. The one at zero energy is presentat any flux and is due to a chiral (bipartite) symmetry[7]. The other two are the result of destructive interfer-ences tuned by the magnetic field. It was demonstratedthat a particle, placed at initial time on one of the threedifferent sites of the lattice, displays a quantum diffusionlimited to a small cluster of sites, and eventually peri-odically bounces back and forth to its original position.This effect disappears if the flux is tuned away from halfa flux quantum [5] or if interactions between particles [6]or disorder [8] are introduced.This predicted phenomenon inspired several experi- ∗ [email protected] † [email protected] ‡ [email protected] mental implementations or proposals, among which su-perconducting wire networks [9], Josephson junction ar-rays [10], cold atomic gases [11], photonic lattices [12],ions micro traps [13], etc...Here, we show that a related caging phenomenon canbe obtained in the a priori different case of unitary quan-tum walks (QW). It displays several novel features thatare absent from the original TB model [5], such as thetuning of the critical flux and that of the cage size.The article is organized as follows. In Section II, weintroduce 1d and 2d discrete-time quantum walks in thepresence of a perpendicular magnetic field. Then, in Sec-tion III, we study AB cages in the case of the 1d QW onthe diamond chain. Next, in Section IV, we obtain ABcages for 2d QW on the T lattice. Finally, in Section V,we conclude with a summary and perspectives. SeveralAppendices provide details of calculations and further ex-amples of AB cages, such as one leading to the tuning ofthe cage’s spatial extension (see Appendix G). II. MAGNETIC QUANTUM WALKS
Quantum walks can be seen as a quantum generaliza-tion of classical random walks (see e.g. Refs. [14, 15]and [16] for review). Their main interest arises in thequantum information framework. We focus on discrete-time QWs. A particle, equipped with a finite dimensionalinternal state (with Hilbert “coin” space H c ), is subjectto unitary shifts along edges on a graph Λ (with Hilbertspace H Λ = span {| i (cid:105) , i ∈ Λ } ), in a direction guided bythe internal state. At each time step, a unitary coin oper-ator C is uniformly applied on the internal states, even-tually affecting the following unitary shift S and leadingto the walk operator W = S C . After N steps, the initialstate | ψ (0) (cid:105) becomes | ψ ( N ) (cid:105) = W N | ψ (0) (cid:105) . QW pro-tocols have already led to several experimental realiza-tions in 1d with trapped atoms [17], single photons [18]or Bose-Einstein condensates in momentum space [19].In 2d, a generalization has been done with light [20], andthe tentative addition of a magnetic field has also beendiscussed [21–24].We consider a QW in a perpendicular magnetic field a r X i v : . [ qu a n t - ph ] J u l and study cage effects on two periodic graphs: (i) theDC with four-fold coordinated “hub” sites a and two-fold“rim” sites ( b, c ) (Fig. 1) (ii) the T tiling with six-foldhub sites a and three-fold rim sites ( b, c ) (Fig. 2). In eachunit cell, the internal space size is 8 for DC and 12 for T .In a standard QW, the internal space codes for the di-rection of the walk. E.g., in 1d, a spin 1 / S is then a sum of hopping terms between the two statesassociated with each edge (one per opposite sites of thisedge) and is therefore unitary and hermitian. The opera-tor C s shuffles the states associated with site s = { a, b, c } and its different incident edges. The coin operator C isthe direct sum of C a , C b and C c . It may be hermitian,but generically does not commute with S , leading to anon trivial walk operator W .We define four types of n × n coins associated withsites of coordination number n = 2 , , ,
6. Previousworks have focused mainly on three types of coins [16]:Hadamard H n , defined for sizes n = 2 m , Grover G n anddiscrete Fourier transform D n coins, which can be con-structed for any matrix size n . As D n ’s were found not tolead to caging effects, we shall not discuss them further.Grover coins read G n = (2 /n ) n − I n , where n is an n × n matrix full of 1 and I n is the identity matrix.For the two-fold sites, we use generic unitary coins: U ( θ, ϕ, ω, β ) = (cid:18) cos θ e i β − sin θ e i( ϕ + ω ) sin θ e − i ω cos θ e i ( ϕ − β ) (cid:19) . The standard Hadamard coin H is U ( π/ , π, ,
0) and SO (2) rotations correspond to U ( θ, , , n = 3,we use SO (3) rotations R ( α, γ ), of angle α around unit3d vectors (cid:126)v = (cid:16) cos γ √ , sin γ, cos γ √ (cid:17) . Such a matrix reducesto G whenever ( α, γ ) = ( π, sin − (1 / √ n = 4, weuse either H = H ⊗ H , or G , and finally G is used for n = 6. The basis conventions are detailed in AppendixA. The shift operator reads S = (cid:88) (cid:104) ( i,j ) , ( i (cid:48) ,j (cid:48) ) (cid:105) | i, j (cid:105) (cid:104) i (cid:48) , j (cid:48) | + h.c. , where i and i (cid:48) are neighbouring sites, and j and j (cid:48) twoopposite directed edges between these two sites. Themagnetic field B = | ∇ × A | enters via a Peierls substitu-tion [26], i.e. the hopping terms in the shift S get multi-plied by a phase factor e i πφ (cid:82) i (cid:48) i dl · A , where A is the vectorpotential and φ = h/e the flux quantum [21, 23, 24]. Be-ing unitary, W can be written as the exponential e − i H eff of an effective hamiltonian H eff , whose evolution is onlyconsidered at integer times. Its eigenvalues are purephases, called quasi-energies and defined modulo 2 π . a bc FIG. 1. Bottom: a piece of a diamond chain with hub sites a (in red) and rim sites b (in green) and c (in blue). An arrowmeans a phase e i πf , which is our gauge choice. The dashedrectangle indicates the maximal extension of a cage at criticalflux, for an initial state localized on the circled a site. Top:the QW Hilbert space is schematized with four (resp. two)basis states for a (resp. b, c ) sites, shown here as circles. Coinsoperate on states inside a circle, and shifts along edges. ab c FIG. 2. Left: a piece of T tiling with hub sites a (in red) andrim sites b (in green) and c (in blue). The dashed hexagonindicates the maximal extension of a cage at critical flux cen-tered on the initial circled a site. Right: the QW Hilbertspace is schematized, with six (resp. three) basis states for a (resp. b, c ) sites, shown here as circles. III. AB CAGES ON THE DIAMOND CHAIN
In contrast to 2d tilings, for the DC (see Fig. 1), itis possible to find a gauge that preserves the periodicityof the lattice (i.e. a unit cell containing 8 basis states).We choose it as a phase e i2 πf on one of the four edges(see bottom of Fig. 1), with the reduced flux f defined asthe magnetic flux per plaquette in units of φ , i.e. f = φ − (cid:72) pl. dl · A . Using translation invariance, it is possibleto diagonalize the W operator into 8 × k -dependentblocks W ( k ) with vanishing diagonal 4 × k is a wave-vector inside the first Brillouin zone.Then, W is made of 4 × i E ( k ) , so that W eigenvalues are e i (cid:15) ( k ) , with (cid:15) = E/ E/ π .Different coins can be used on the b and c sites. Forexample, with C a = G , C b = U ( θ, ϕ, , β ) and C c = U ( θ, ϕ, ω, β ), one gets the following “quasi-energies” (cid:15) ( k ), leading to four β -independent bands (cid:15) ( k ) and fourflat bands (cid:15) fb : (cid:15) ( k ) = ϕ + π ±
12 cos − (cid:104) sin θ cos (cid:16) πf − ω (cid:17) × cos (cid:18) πf − ω k + π − ϕ (cid:19)(cid:21) ± π (cid:15) fb = ± π ϕ ±
12 cos − (cos( β − ϕ/
2) cos θ ) . The quasi-energies versus flux patterns display sym-metries: two translations, f ←→ f (due to thePeierls substitution) and (cid:15) ←→ (cid:15) + π (due to a bi-partitegraph [23]) and two mirrors, f ←→ − f + ω/π (with k ←→ − k − π + ϕ ) and (cid:15) ←→ − (cid:15) + ϕ/ k ←→ k + π ).Fig. 3-a,b show the resulting W quasi-energies displayedbetween 0 and π owing to the above translation symme-try, for C a = G and symmetric coins ( C b = C c , meaningthat ω = 0). They display two striking features: (i) theexistence of flat bands (versus k and f , which are even θ -independent whenever β − ϕ/ ± π/ f c at whichthe pinching occurs can be tuned at will, if ω (cid:54) = 0 (there-fore C b (cid:54) = C c ), as f c = 1 / ω/ π , and even made tovanish (not shown in the Figure). FIG. 3. Diamond chain quasi-energies spectra, in the range[0 , π ] (repeated under a π translation), plotted versus the re-duced flux f (invariant under translation of period 1), forthe coins { C a , C b = C c } equal to: (a) { G , U ( π/ , π, , } ,(b) { G , U ( π/ , , , } , (c) { H , U ( π/ , π, , π ) } , (d) { H , U ( π/ , , , } . Varying ω would lead to an horizontalshift of the whole pattern. The critical flux [either 1/2 for (a)and (b) or 0 ∼ These two features (i) and (ii) are independent: indeed,using a H coin on a sites (instead of G ) still leads toa pinching, but without flat bands (Fig. 3- c, d ). Here f c = ω/ π and pinching occurs at vanishing magneticfield whenever identical coins are applied on sites b and c (i.e. ω = 0). At f c the 8 quasi-energies read: (cid:15) = π ± π ϕ ± × cos − sin (cid:0) β − ϕ (cid:1) cos θ ± (cid:113) θ + cos θ sin ( β − ϕ )2 The pinching is associated with an AB-like caging ef-fect. In the TB case [5], the AB caging was proved byanalyzing the local density of states, with a Lanczos tridi-agonalization showing a vanishing recursion coefficientfor f c = 1 / W is transformed into an almost tri-angular matrix with an added subdiagonal (Hessenbergform) using the Arnoldi iteration (see Appendix C andRef. [27]). The latter consists in starting from a givenstate | (cid:105) and then iteratively applying W . Each obtainednew state | n + 1 (cid:105) is made orthogonal to all previous ones {| m (cid:105) , m ≤ n } according to: b n +1 | n + 1 (cid:105) = W | n (cid:105) − (cid:88) m (cid:54) n (cid:104) m | W | n (cid:105) | m (cid:105) . A vanishing coefficient b n with n = n c terminates theiteration and indicates a caging effect. Indeed, startingfrom a localized state, this proves that only a finite frac-tion of states can be reached, leading to an evolutioninside the subspace spanned by the states {| n (cid:105) , n ≤ n c } .A vanishing coefficient b is indeed found for all abovedescribed DC cases, with an initial state localized at anyinternal state on an a site, see Fig. 4-a. This leads to acage of maximal radius twice the size of the unit cell (seeFig. 1). FIG. 4. Arnoldi recursion coefficients: a) DC case with a G coin: plot of b versus ( f, ϕ ) for β = 0 and θ = π/ , π/ , π/ T case: plot of b versus ( f, γ ) for α = π/ , π/ , π/
6. Thecritical flux f c = 1 / A last point concerns the time evolution inside a cage.The above quasi-energies are incommensurate for generic( θ, ϕ, β ) values, which leads to a quasiperiodic time be-havior. A periodic behavior can nevertheless arise as dis-played in Table I. A sketch of the dynamics of a period8 cage is shown in Appendix D.
Graph C a C c FB f c PeriodDC G U ( θ, ϕ, ω, β ) generic Yes + ω π QPDC G U ( θ, ϕ, ω, ± π + ϕ ) Yes + ω π G U ( πpq , ϕ, ω, − ϕ ) Yes + ω π LCM(4 , q )DC H U ( θ, ϕ, ω, β ) generic No ω π QPDC H U ( π , π, ω,
0) No ω π H U ( π , π , ω,
0) No ω π H U ( π , , ω,
0) No ω π T G R ( α, γ ) generic Yes QP T G ˜ R ( α, γ, − π ) Yes QP T G R ( π , γ ) Yes T lattice for different coins C a and C c [with C b = C c ( ω =0)]. The table indicates the critical flux f c for the spectralpinching, FB tells whether flat bands are present, and thelast column specifies the period of the time evolution when itexists (QP referring to a quasiperiodic case). IV. AB CAGES ON THE T LATTICE
Next, we consider a 2d QW on the T lattice (seeFig. 2). Using the G coin on a sites, we found suitable R ( α, γ ) coins for ( b, c ) sites that lead to caging. As inthe DC case, we allow for a different R coin operatingon the c and b sites by introducing an angle ω enter-ing ˜ R ( α, γ, ω ) i,j = R ( α, γ ) i,j e − iω (cid:80) k ε ijk with ε ijk theskew-symmetric Levi-Civita tensor. This unitary coin isno longer a 3d rotation. As in the DC, the introductionof ω (cid:54) = 0 breaks time-reversal symmetry. Using a Landaugauge, periodicity is present in the (say) y direction (witha magnetic unit cell containing 12 q states), and W can bediagonalized for rational f = p/q into 12 q × q blocks,leading to a Floquet-Hofstadter butterfly [28] shown inFig. 5. It displays flat bands, apparent self-similar sub-patterns and pseudo-Landau levels. Here, we focus on thespectral pinching that occurs near f c = 1 /
2. The quasi-energies, doubly degenerate and k -independent, read: (cid:15) = 0 , ± π , π, π ± α , − π ± α ,π ±
12 cos − (cid:18) α (cid:19) , − π ±
12 cos − (cid:18) α (cid:19) . These expressions show that the dynamics of the QWcages can be periodic or not depending on whether quasi-energies differences are commensurate or not (see Ap-pendix E).Using the Arnoldi iteration, one numerically finds thatthe b coefficient vanishes at f c = 1 /
2, as shown inFig. 4-b. The associated AB cage depends on the precisechosen initial state, the largest one, displayed in Fig. 2,being larger than in the TB case.The flux values ± / W can be diagonalized into 12 ×
12 blocks W ( k x , k y ). Asymmetric ˜ R coins can then change the critical f c value: when ω = − π/ f c is shifted from 1 / /
6, according to f c = 1 / ω/ π . In contrast to theDC case, here f c cannot be tuned continuously. FIG. 5. Floquet-Hofstadter butterfly on the T lattice (i.e.quasi-energy spectra of a QW versus magnetic flux f ) forthree values of the R coin operator. Top: α = 2 π/ , γ =sin − (1 / √ , ω = 0, Middle: α = 2 π/ , γ = 0 , ω = 0, Bottom: α = π/ , γ = sin − (1 / √ , ω = − π/
3. The vertical dashedlines indicate the critical flux f c = 1 / V. CONCLUSION
As in the TB case, the AB cages in a QW result from adestructive interference tuned by the magnetic field en-tering through Peierls phases. However, in the presentcase, the action of the coin operator on the internal de-grees of freedom is crucial for the caging to occur. Ampli-tudes vanish on those states that would otherwise allowthe particle, upon further shift, to escape the cage. Asa result, the AB cages in the QW case have a larger sizethan their TB counterparts.Playing with different type of coins, cages can occurat critical fluxes others than 1 /
2. Necessary conditionsfor this tuning of f c are the existence both of a gaugerespecting the translation symmetries of the structure,and of a coin that breaks time-reversal symmetry (corre-sponding to ω (cid:54) = 0 in the rim coin operators) and partiallycompensates the effect of the applied magnetic field. Forexample, in the DC, the G (resp. H ) coin creates ABcages at flux f c = 1 / f c . Therefore, in DC, a periodic substitutionby one type of coin (say G in an array of H ) allows oneto engineer AB cages (here at f c = 1 /
2) of arbitrary sizeby tuning the G coins inter-distance (see Appendix G).The QW AB cages and the corresponding Floquet-Hofstadter butterfly should be readily observable. Re-cently, a concrete experimental implementation of a mag-netic QW in 2d has already been proposed [23]. Theextra challenge is to implement such a QW on a latticewith varying coordination number. QW on such latticeshave been theoretically studied for many years [25], butwe are not aware of an efficient experimental realizationto date. A brute force solution would be to have a largeinternal space (8 for DC or 12 for T ). Finally, a strikingconfirmation of the above predictions would be to testthe tunability of the critical flux at which caging occursand that of the cage size.Although QW may be seen as a subclass ofperiodically-driven Hamiltonian systems, our proposalfor the DC is significantly different from the AB cagesseen in [12] that uses Floquet engineering to realize theTB model of [6] and does not involve the action of coinsin an internal space. ACKNOWLEDGMENTS
We thank Janos Asb´oth for useful discussions.
Appendix A: Basis conventions
The differents coin operators, acting at a given site,are given in the main text. We add here our chosen basisordering convention in the different cases, relative to thematrix representation of different coins.
FIG. 6. Basis states ordering convention for coordinated sites.(Top) diamond chain with two 2-fold (rim b, c ) and one 4-fold(hub a ) sites. (Bottom) T tiling with two 3-fold (rim b, c )and one 6-fold (hub a ) sites. On the diamond chain, the right (resp. left) inter-nal state of sites b and c corresponds to the first (resp.second) vector in the matrix representation: (cid:32) (cid:33) . The action of the coin is the multiplication by the generic uni-tary operator U ( α, ϕ, ω, β ). For sites a , the first vector isthe state on the right-up side of the site, and the basis isanti-clockwise oriented. For instance, a quantum walkerlocalized on the left down internal state of a 4-fold site,indicated by the number 3 on top of Fig. 6 is representedby the vector: . Then, the application of the Groveror the Hadamard matrix defines the action of the coin. Appendix B: Isospectrality of W sub-blocks In the two examples treated here, the underlying graphis bipartite. The Hilbert space basis is the union of twoparts B a and B b,c , and the action of the unitary operator W is bipartite: it sends components in B a onto compo-nents in B b,c and vice versa. As a consequence W sendseach sub-basis onto itself, and appears therefore block-diagonal.Now, suppose that | ψ (cid:105) = | ψ a (cid:105) + | ψ b,c (cid:105) is an eigenvectorof W , with eigenvalue e i (cid:15) , with | ψ a (cid:105) (resp. | ψ b,c (cid:105) ) beingthe subpart of | ψ (cid:105) in B a (resp. B b,c ). | ψ (cid:105) is an eigenvectorof W , with eigenvalue e (cid:15) . In each diagonal sub-blocksof W , spanned by B a and B b,c , we therefore have that | ψ a (cid:105) and | ψ b,c (cid:105) are separately eigenvectors of W withthe same eigenvalue e (cid:15) , the two blocks being thereforeisospectral. Appendix C: Arnoldi iteration
The Arnoldi algorithm [27] amounts to tranform ageneric square matrix into a so-called “Hessenberg form”,which is an almost triangular form. More precisely, itresults to a (say upper) triangular form plus the lowersub-diagonal.Here we apply the Arnoldi iteration to reduce the uni-tary QW operator. We start with a localized state | (cid:105) (the initial condition stating where the wavefunction isconcentrated at t = 0) and successively apply the QWoperator W . At each step, a new state is obtained fromwhich the part which belongs to states already exploredis removed: b n +1 | n + 1 (cid:105) = ˆ W | n (cid:105) − (cid:88) m (cid:54) n (cid:104) m | W | n (cid:105) | m (cid:105) . As a consequence, a new orthonormal basis | n (cid:105) is gen-erated in which ˆ W has an Hessenberg form. Note that,when the operator is hermitian, the Arnoldi algorithmreduces to the Lanczos tridiagonalization.The b n are defined positive, and the algorithm endswhenever a new generated state is null ( b j = 0). As aconsequence, starting from the (localized) state | (cid:105) , thequantum walk explores only a finite number of states, leading to a cage trapping. Appendix D: Sketch of the cage effect in a simple DC case
Fig. 7 shows a sketch of a periodic quantum walk on the diamond chain, at the critical value corresponding to acaging effect. We take the simple example of a period 8 quantum walk, with parameters given in the caption.
FIG. 7. Sketch of the dynamics a period 8 quantum walk on the diamond chain with coins C a = G , C b = C c = U ( π/ , , , − π/
2) and f c =1/2. The probability to find the walker on each eigenbasis state is shown as a function of in-teger time T . Appendix E: Criteria for periodic dynamics at thecaging critical value f c At the critical caging value f c , an interesting ques-tion is to analyse the confined dynamics. It is periodicwhenever the differences between quasi-energies are com-mensurate. In the tight binding case the dynamics werefound to be periodic in all cases (DC and T ). QW cagingeffects are found here with a larger set of parameters andare generically quasiperiodic. However, the analysis ofthe quasi-energies in the expressions given in the maintext allows one to find some conditions for periodicity.
1. DC case
The fact that caging effects are found with a generic U (2) coin U ( θ, ϕ, ω, β ) leads to a large spectrumof parameters. In the case C = G , one findsfor instance that the dynamics is periodic whenever cos − (cos( β − ϕ/
2) cos θ ) is commensurate with π . Forinstance, if β − ϕ/ , q ), whenever θ = 2 πp/q with ( p, q ) ∈ Z .On the other hand, if β − ϕ/ ± π/
2, the resultingdynamics has a period 8 for any θ . T case Substracting the quasi-energies at the critical f c value,we have that α must be commensurate with π , say α = πp /q and satisfy an additional more compli-cate equation: cos − (cid:16) (2 + cos πp q ) / (cid:17) = π p q . If thisis verified, the quantum walk is periodic, with period4 × LCM( q , q ). We find that for α = 2 π/
3, the dynam-ics has period 12, and numerically that no other solutionexists for q , q ≤ Appendix F: Example of a periodic gauge at flux-1/3 for the T tiling. e i π e i π e i π e i π e i π e i π u u FIG. 8. An example of a periodic gauge at flux − / T tiling Fig. 8 shows an example of a gauge at flux -1/3 sharingthe T tiling periodicity (whose unit vectors are displayedin red). Appendix G: Cages with tunable size in the DC case
In the DC case, Grover G and Hadamard H coins onthe a hub sites create cages at different fluxes. Playingwith this new feature allows one to control the extensionof the cage. We briefly describe an example of that pro-cedure. Start, at flux 1 /
2, with a chain of H coins onthe hub sites, which do not therefore create cages (seeFig. 3), and substitute some H by G coins in a peri-odic manner. This induces AB cages, whose size dependsboth on the G − G distance (the superlattice period),and on the initial state of the quantum walk.The rules to determine the extension of a cage startingfrom an initial state localized on a site a , in the DC unitcell labelled n , is: • First rule : The coin on the initial site n is irrele-vant for the cage to occur. What counts are coinsapplied on neighbouring sites. • Second rule : On the right-hand side, the QWspreads until it meets a substitution coin which willstop it on the right next site. • Third rule : On the left-hand side, the coin on thefirst neighbour of the initial site does not matter.Then, the QW spreads and stops exactly when itmeets a substitution coin.Fig. 9 gives an example, at flux 1 /
2, for a G coininserted every 5 hub sites (in a chain containing only H coins otherwise). Cages are represented by dashed boxeswhere the color indicates the location of the initial state.To understand the above rules, it is useful to operate abasis change within the internal space at the hub sites, G H H H H G H H H H G H H H H G H H H H